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Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities: The Inverse Problem
Author(s) -
Dagan Gedeon
Publication year - 1985
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr021i001p00065
Subject(s) - mathematics , covariance , covariance function , inverse problem , probability density function , conditional probability distribution , hydraulic head , flow (mathematics) , variogram , groundwater flow , truncation (statistics) , mathematical analysis , statistics , aquifer , geometry , groundwater , geology , geotechnical engineering , kriging
The inverse problem is defined here as follows: determine the transmissivity at varius points, given the shape and boundary of the aquifer and recharge intensity and given a set of measured log‐transmissivity Y and head H values at a few points. The log‐transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function (pdf). The solution of the inverse problem is the conditional normal pdf of Y , conditioned on measured H and Y , which is expressed in terms of the unconditional joint pdf of Y and H . The problem is reduced to determining the unconditional head‐log‐transmissivity covariance and head variogram for a selected Y covariance which depends on a few unknown parameters. This is achieved by solving a first‐order approximation of the flow equations. The method is illustrated for an exponential Y covariance, and the effect of head and transmissivity measurements upon the reduction of uncertainty of Y is investigated systematically. It is shown that measurement of H has a lesser impact than those of Y , but a judicious combination may lead to significant reduction of the predicted variance of Y . Possible applications to real aquifers are outlined.